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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2506.19443 |
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| _version_ | 1866908496117628928 |
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| author | Li, Jian-Rong Tewari, Ayush Kumar |
| author_facet | Li, Jian-Rong Tewari, Ayush Kumar |
| contents | The Grassmannian cluster algebra $\mathbb{C}[\text{Gr}(k, n)]$ admits a distinguished basis known as the dual canonical basis, whose elements correspond to rectangular semi-standard Young tableaux with $k$ rows and with entries in $[n]$. We establish that each such tableau induces a positroidal subdivision of the hypersimplex $Δ(k,n)$ via a map introduced by Speyer and Williams. For $\text{Gr}(2,n)$, we prove that non-frozen prime tableaux correspond precisely to the coarsest positroidal subdivisions of $Δ(2,n)$. Furthermore, we present computational evidence extending these results to $k>2$. In the process, we formulate a conjectural formula for the number of split positroidal subdivisions of $Δ(k,n)$ for any $k \ge 2$ and explore the deep connections between the polyhedral combinatorics of $Δ(k,n)$ and the dual canonical basis of $\mathbb{C}[\text{Gr}(k, n)]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19443 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From dual canonical bases to positroidal subdivisions Li, Jian-Rong Tewari, Ayush Kumar Combinatorics Quantum Algebra 52B11, 52B40, 05E10, 14M15, 13F60 The Grassmannian cluster algebra $\mathbb{C}[\text{Gr}(k, n)]$ admits a distinguished basis known as the dual canonical basis, whose elements correspond to rectangular semi-standard Young tableaux with $k$ rows and with entries in $[n]$. We establish that each such tableau induces a positroidal subdivision of the hypersimplex $Δ(k,n)$ via a map introduced by Speyer and Williams. For $\text{Gr}(2,n)$, we prove that non-frozen prime tableaux correspond precisely to the coarsest positroidal subdivisions of $Δ(2,n)$. Furthermore, we present computational evidence extending these results to $k>2$. In the process, we formulate a conjectural formula for the number of split positroidal subdivisions of $Δ(k,n)$ for any $k \ge 2$ and explore the deep connections between the polyhedral combinatorics of $Δ(k,n)$ and the dual canonical basis of $\mathbb{C}[\text{Gr}(k, n)]$. |
| title | From dual canonical bases to positroidal subdivisions |
| topic | Combinatorics Quantum Algebra 52B11, 52B40, 05E10, 14M15, 13F60 |
| url | https://arxiv.org/abs/2506.19443 |